6.5 Curve Sketching
6.5.1 Increasing and Decreasing Functions and The First
Derivative Test
LEARNING OUTCOMES:
A function f is increasing on an interval I if:
x1 , x2 I, x1 < x2 f (x1 ) < f (x2 )
On completion of this section you should
1. be able to use

We can see that in particular Rolles Theorem implies that if a dierentiable
curve crosses the x-axis twice ( f (a) = f (b) = 0), there must be a point
between successive crossings where the tangent to the curve is parallel to
the x-axis.
Rolles Theorem re

y
Local max
2
POI
1
POI
1
x
POI
Local min
2
The graph above does not depict all features and is only a rough sketch. The
intercepts are missing. This leads to the culmination of the applications of
the derivative into the procedure to sketch a curve showi

Worked Example 6.2.3 (Inclination, Trigonometry). A balloon leaves the
ground 50m away from an observer and rises vertically at a rate of 14m/min.
At what rate is the angle of inclination of the observers line of sight increasing at the instant when the b

Theorem 6.3 (Fermats Theorem). If has a local maximum or minimum at Worked Example 6.3.3 (Converse of Fermats Theorem is false). Prove
an interior point c of dom( f ) and if f is dierentiable at c, then f (c) = 0. that f (x) = x3 has no local extremum.
P

Worked Example 6.5.3. Use the First Derivative Test to nd the local ex- 6.5.2 Concavity and The Second Derivative Test
1
trema of g(x) = (x4 4x3 ).
You have seen that the rst derivative locates the intervals of increase and
9
decrease of a function f by p

University ot the Witwatersrand, Johannesburg
Schooi of Physics
PHYSIOQi/ LABORATORY TEST 1 15 May 2008
40 marks 40 minutes
ENSTRUCTIONS
Answer ali six Questions on the answer book provided Ensure that you ll in your name,
surname and student number in

Torque or Leverage
The Physics of See Saws . .
. . & other stuff
Moments
Couples
Equilibrium
Conditions for Equilibrium
Centre of Gravity
Equilibrium of Bodies
The moment of a force about an
axis is defined as the product of the
force and the perpend

University of the Witwatersrand, Johannesburg
School of Physics
Physics I (PHYS 1001/1006)
Lecture Examples: 1.1 continued : Scalars and Vectors
Year : 2016
For additional information see the tutorial data sheet.
1.
Four velocity vectors are shown below:

Dynamics the study of forces and their
effect on motion.
Force and Momentum
Newtons Laws of Motion
Newtons Law of Gravitation
Dynamics of Circular Motion
Friction
Force is defined as any interaction which
tends to change the motion of an object.
Forc

Review
Elasticity
Stress and strain
Tensile and shear stress and strain
Hookes law, elastic modulii
Stress-strain curve
2.1/4
Discussion
Which is more elastic steel or rubber and why?
Hydrostatics
Pressure difference between different levels
of a liq

Lecture Example 7
The acceleration due to gravity at the surface of planet
X is twice that on the surface of Earth. The radius of
X is also twice the radius of Earth.
(a) Show that the mass of planet X equals eight times
the mass of Earth.
(b)Use a first